Can you give more information on the problem? It could simply be that you've take a particular direction to be positive, and the answer in the back of the book assumes the opposite direction is positive.

The integral you posted first is not correct as cynapse explained, even in corrected form it's an indefinite integral with the constant of integration missing. It sounds like you need a definite integral.
Did you get the right definite integral for the limits of integration 0 and T?
The answer for the equation as you described it is -0.0084.
This leads me to believe that you don't have the correct set up for the problem.
Are the units consistent?
Do you have the correct limits of integration?
If you post the full wording of the problem as well as units of known quantity's with a more complete description of what you are doing I can try to help you more.

Everything is correct, we are setting up a spreadsheet in class. We use this for Analytical
approach. v is calculated using the equation above, so to get back to y, you have to integrate. I know what y should be which I stated above because that's through the numerical approach.

It seems you're finding distance travelled while experiencing air resistance. Just do this:
[tex]\frac{mg}{b}\int(1-e^{-bt/m})dt[/tex]
Then, integrate each term in the integral separately in respect to time t (find the integral of 1 and the integral of -e^(-bt/m) individually). For the integral of -e^(-bt/m), use (-bt/m) as a function u. Then simply use the chain rule for integration to solve the problem.

That's the correct integral for the limits 0 & T.
At least I think it is. I would recommend using the forums handy mathematical formatting so the equations are easier to read.
So I still think something else is wrong here.